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Arch. Hist. Exact Sci. (2013) 67:149–170DOI
10.1007/s00407-012-0107-8
Hendrik Antoon Lorentz’s struggle with quantumtheory
A. J. Kox
Received: 15 June 2012 / Published online: 24 July 2012© The
Author(s) 2012. This article is published with open access at
Springerlink.com
Abstract A historical overview is given of the contributions of
Hendrik AntoonLorentz in quantum theory. Although especially his
early work is valuable, the mainimportance of Lorentz’s work lies
in the conceptual clarifications he provided and inhis critique of
the foundations of quantum theory.
1 Introduction
The Dutch physicist Hendrik Antoon Lorentz (1853–1928) is
generally viewed as anicon of classical, nineteenth-century
physics—indeed, as one of the last masters ofthat era. Thus, it may
come as a bit of a surprise that he also made important
contribu-tions to quantum theory, the quintessential non-classical
twentieth-century develop-ment in physics. The importance of
Lorentz’s work lies not so much in his concretecontributions to the
actual physics—although some of his early work was
ground-breaking—but rather in the conceptual clarifications he
provided and his critique ofthe foundations and interpretations of
the new ideas. Especially in his correspondencewith colleagues,
such as Max Planck, Wilhelm Wien and Albert Einstein, he time
andagain tried to clarify the quantum principles and explore their
consequences.
In this paper I will give an overview of Lorentz’s work in
quantum theory, includ-ing his informal contributions in
discussions and correspondence. I will first discussLorentz’s early
work on radiation theory, in which he gives a derivation of a
radiationlaw from classical electron theory. Then I will discuss
Lorentz’s 1908 Rome lecture
Communicated by : Jed Buchwald.
A. J. Kox (B)Institute for Theoretical Physics, University of
Amsterdam,Amsterdam, The Netherlandse-mail: [email protected]
123
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150 A. J. Kox
in some technical detail, because the theory Lorentz develops in
the lecture is a daringnew application of Gibbs’s statistical
mechanics and because its outcome is that themost general classical
theory one can think of leads inescapably to the
Rayleigh–Jeansradiation law. A key element in this derivation is
the theorem of equipartition of energy.
The Rome lecture was widely discussed, both in correspondence
and in papers. Aswe will see, in the discussions with Lorentz and
also in his later work three questionswill keep coming back. The
first one is the validity of the law of equipartition of energyand
its possible modifications. A second theme, how to deal with the
quantum discon-tinuity (also discussed in the lecture, although
less explicit), would soon evolve in thespecific question of where
to localize the discontinuity: in the ether, in the
interactionbetween matter and ether, in the resonators or
elsewhere. A third theme, not explicitlypresent in the Rome
lecture, but gaining much prominence in later work, is the matterof
the light quantum. Do independent light quanta exist, and if so,
how to reconciletheir existence with typical wave phenomena such as
interference? This last theme isin particular discussed in
correspondence with Einstein.
The paper concludes with a discussion of the way Lorentz dealt
with the paralleldevelopments of matrix mechanics and wave
mechanics. He carefully studied boththeories but engaged with them
in totally different ways: he extensively discussedand criticized
wave mechanics, especially in correspondence with Erwin
Schrödinger,whereas in the case of matrix mechanics he limited
himself to trying to master the for-malism and grasping its
consequences. Although he had much respect and admirationfor the
work of the younger generation, he remained critical and to the end
couchedhis quantum work in classical terms.
2 Early work on radiation theory
Lorentz’s work on radiation theory is characterized by the same
methodological con-sistency that we find throughout his work and
that culminated in his mature theory ofelectrons in the first
decade of the twentieth century. He bases himself on an ontologyof
particles and fields, or, to use his own terminology, on a strict
separation of mat-ter—consisting of charged and neutral
particles—and ether. The latter acts as carrierof the
electromagnetic action, caused by the presence of charged
particles. Ether andmatter are separate entities, that act on each
other, but must be treated differently. Theparticles obey the laws
of classical mechanics; the ether is governed by
Maxwell’sequations.1
The first paper I want to discuss dates from 1903, and has the
title: “On the emis-sion and absorption by metals of rays of heat
of great wave-lengths” (Lorentz 1903).Lorentz bases himself on the
theories of Paul Drude and Eduard Riecke for the electri-cal
conductivity of metals, in which it is assumed that metals contain
large quantitiesof freely moving electrons, that regularly collide
with the metal atoms. Applying hiselectron theory to this model,
Lorentz calculates the energy density of radiation emit-ted by the
electrons by assuming that they only radiate when colliding with
atoms,
1 In Lorentz’s early work the ether is still treated as a
mechanical system, but gradually it loses all of itsmechanical
properties save one: its immobility. See Kox (1980) for further
details.
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Hendrik Antoon Lorentz’s struggle with quantum theory 151
limiting himself to the case of long wavelengths. In a dazzling
show of mathematicalprowess he finds that the radiation is
distributed according to the Rayleigh–Jeans law(which, of course,
was not yet known by that name in 1903) and also that it conformsto
the long wavelength limit of Planck’s brand-new radiation law:
f (λ, T ) = 8πkTλ4
. (1)
In this paper Lorentz makes his first comments on Planck’s
quantum hypothesis.He writes:
[…] the hypothesis regarding the finite “units of energy”, which
has led to theintroduction of the constant h, is an essential part
of the theory; also that thequestion as to the mechanism by which
the heat of a body produces electromag-netic vibrations in the
aether is still left open. Nevertheless, the results of Planckare
most remarkable.
And later on, when comparing his result with Planck’s work, he
comments:
There appears therefore to be a full agreement between the two
theories inthe case of long waves, certainly a remarkable
conclusion, as the fundamentalassumptions are widely different.
It is interesting to note that in his paper Lorentz
characterizes Planck’s theory in thefollowing way:
It will suffice to mention an assumption that is made about the
quantities ofenergy that may be gained or lost by the resonators.
These quantities are sup-posed to be made up of a certain number of
finite portions, whose amount isfixed for every resonator;
according to Planck, the energy that is stored up ina resonator
cannot increase or diminish by gradual changes, but only by
whole“units of energy”, as we may call the portions we have just
spoken of.
According to Thomas Kuhn, this is an “anomalous” reading of
Planck’s work, thatLorentz corrected later.2 Closer study of
Lorentz’s published and unpublished utter-ances shows, however,
that Lorentz never strayed very far from this interpretation.For
instance, in his 1908 Rome lecture (see Sect. 3), he uses almost
identical words,whereas Kuhn uses a quotation from a letter from
Lorentz to Wilhelm Wien (to theeffect that processes in the ether
take place in a continuous way), which was writtena bit after the
lecture, to argue that Lorentz had dropped his earlier
interpretation.3
What Lorentz was uncertain about, as will be shown in Sect. 6
below, was where thediscontinuity lay: in the interaction between
ether and matter (i.e., the resonators) orin the interaction
between the resonators and the other matter.
2 See Kuhn (1978, p. 138).3 See Kuhn (1978, p. 194). The letter
in question is dated 6 June 1908; it is reproduced in Kox (2008)
asLetter 171. See also Sect. 4.1 below for a further discussion of
the Lorentz–Wien correspondence.
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152 A. J. Kox
3 The Rome lecture
From Lorentz’s later work on radiation theory, it becomes clear
that the paper discussedin the previous section sets the
methodological stage for later developments along sim-ilar lines
and that its critical tone about the mystery of the underlying
mechanism inPlanck’s theory of energy elements remains an important
theme.
The first important one of the later papers is the 1908 Rome
lecture. It was deliveredat a somewhat strange venue, namely, the
4th International Congress of Mathemati-cians in April 1908. Its
impact has been considerable, if only because it made indis-putably
clear that Planck’s energy elements were fundamentally foreign to
classicalmechanics and electrodynamics.
The lecture, entitled “The distribution of energy between
ponderable matter andether”,4 starts with a lengthy and very
general discussion of the foundations of mechan-ics, kinetic gas
theory, and electrodynamics, clearly meant for an audience of
non-physicists. It is useful to follow Lorentz’s reasoning in some
detail, because it bringsout the systematic way he has set up this
paper.
Lorentz first reviews the basics of radiation theory. From
Kirchhoff’s work it followsthat a universal radiation law exists:
the energy density of radiation in the wavelengthinterval λ, λ+ dλ
at temperature T is given by:
F(λ, T ) dλ, (2)
with F a function that is independent of the specific properties
of the body that hasproduced the radiation.
Next Lorentz discusses the law of equipartition of energy,
Boltzmann’s work inkinetic gas theory, and the at that time not
widely known statistical mechanics of Gibbs,as an alternative to
Boltzmann’s approach. He stresses that the phase space
approachtaken by Gibbs only works if the system under consideration
can be described usingHamilton’s equations, so that the ensemble
behaves like an incompressible fluid (inother words, if Liouville’s
theorem holds for the ensemble density). He then introducesthe
ensemble density of the canonical ensemble in the form:
ϕ = Ce−E/Θ (3)
and explains how one can determine macroscopic quantities from
ensemble averages.Lorentz now turns to electromagnetism. He argues
that one needs to base this theory
on a variational principle of the form:
δ
∫(L − U ) dt = 0 (4)
4 The lecture was published in different versions and in various
places; see Lorentz (1908a) in the bibliog-raphy for more
details.
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Hendrik Antoon Lorentz’s struggle with quantum theory 153
with L the magnetic energy and U the electric energy. This
becomes the standardLagrangian for the electromagnetic field if one
takes:
L = 12
∫H2 dV (5)
and
U = 12
∫D2 dV (6)
where D is the dielectric displacement and H the magnetic force.
Note that, althoughthe integrand in (3) has the form of a
“standard” lagrangian, in this case the terms Land U do not have
their usual meanings of kinetic and potential energy,
respectively.It will become clear in the following why Lorentz is
using this suggestive notation.
Having set the stage, Lorentz proceeds to consider the most
general physical sys-tem he can think of. It consists of charged
particles (‘electrons’), neutral particles(‘atoms’), and ether
(i.e., electric and magnetic fields), enclosed in a rectangular
box.The particles are all in motion; the electrons may be free or
bound inside of atoms.This system can be described by four sets of
generalized coordinates:
– {q1} for the neutral particles,– {q2} for the charged
particles,– {q3, q ′3} for the electric field.While {q1} and {q2}
have a straightforward meaning, the ‘coordinates’ {q3, q ′3} have
amore complicated interpretation. For each instant of time one can
split the electric fieldinto two parts: the first is the field that
would exist if all charged particles were at restat their positions
q2, while the second part obeys the source-free Maxwell equation∇ ·
D = 0. This latter part can be expanded in a Fourier series of the
modes that fit inthe box; {q3, q ′3} are the coefficients appearing
in this expansion. Thus, the the threecomponents of D can be
written as:
Dx =∑
u,v,w
(q3α + q ′3α′) cosuπ
fx sin
vπ
gy sin
wπ
hz
Dy =∑
u,v,w
(q3β + q ′3β ′) sinuπ
fx cos
vπ
gy sin
wπ
hz (7)
Dx =∑
u,v,w
(q3γ + q ′3γ ′) sinuπ
fx sin
vπ
gy cos
wπ
hz.
Here f, g, h are the lengths of the sides of the box and u, v, w
are integers running from1 to ∞.5 For each component and for each
set {u, v, w} three directions are defined,perpendicular to each
other. One is determined by the vector ( uf ,
vg ,
wh ), and the other
two, corresponding to the two polarization states of the field,
have the direction cosines
5 The particular choice of sines and cosines is dictated by the
boundary conditions in the box: the sides ofthe box are supposed to
be perfectly conducting, which means that D is always perpendicular
to the sides.
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154 A. J. Kox
α and α′, β and β ′, and γ and γ ′, respectively. The
coefficients {q3, q ′3} (which, fol-lowing Lorentz’s usage, will be
abbreviated to {q3} in the following) obviously dependon {u, v, w}
and on t .
The next step is to write the Lagrangian L − U as a function of
the generalizedcoordinates. Because Lorentz now considers a more
general system than just electro-magnetic fields, he includes
kinetic energy terms in L and potential energy terms inU , in
addition to the field terms. For U he finds
U = U0 + 116
f gh∑
q23 (8)
with U0 a function of the coordinates q1 and q2, accounting for
the potential energybetween the particles; the second term is
12
∫D2dV .
The term L consists in the first place of a part representing
the kinetic energy ofthe electrons and of the neutral particles,
denoted by L0 and quadratic in q̇1 and q̇2.Further, to find the
field part 12
∫H2dV it has to be taken into account that magnetic
fields are generated by moving charges as well as by changing
electric fields. Thisgives rise to terms in H proportional to q̇2
and to q̇3, respectively, which leads to termsin L proportional to
q̇22 , q̇
23 , and q̇2q̇3. The terms proportional to q̇
22 are absorbed in
L0; the explicit form of the part proportional to q̇23 is found
by first calculating Ḋ from(7) and then using ∇ ∧ H = (1/c)Ḋ to
find H. The final result is:
L = L0 + f hg16c2
∑ q̇32π2(u2/ f 2 + v2/g2 + w2/h2) +
∑i j
li j ˙q2i ˙q3 j . (9)
In the last term the index i refers to the individual electrons;
the index j abbrevi-ates the dependence of the quantities q̇3 on u,
v, w and the direction cosines α, β, γ .Accordingly, the
coefficients li j depend on u, v, w, α, β, γ and on the coordinates
q2i .
From this Lagrangian the Lagrange equations follow easily.
Lorentz explicitlywrites down the equations for q3 and shows that
they give rise to standing waves,with wavelength
λ = 2√u2/ f 2 + v2/g2 + w2/h2 (10)
Whenever there is a valid Lagrangian formalism, one can also
write down Hamilto-nian equations. That does not mean that Gibbsian
statistical mechanics can be appliedto the system under
consideration yet. A major obstacle remains: because there
areinfinitely many terms in the sum (7) there are also infinitely
many coordinates q3, sothat phase space would become infinitely
dimensional, precluding the existence of ameaningful ensemble
density. Lorentz’s workround solution is to introduce what hecalls
“fictitious connections” (“liaisons fictives”) in the ether by
which standing waveswith smaller wavelengths than some value λ0 are
excluded. As can be seen from (10)this means that an upper limit is
imposed on the values taken by u, v, w. Lorentzjustifies his
condition by pointing out that one can make λ0 as small as one
wishes. Tothis “fictitious” system Gibbsian statistical mechanics
is then applied.
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Hendrik Antoon Lorentz’s struggle with quantum theory 155
Lorentz now returns to expression (3) for the ensemble density.
He first notes that inL the term L0 contains terms of the form
(1/2)mq̇12. Because E = L + U , the expo-nential in (3) contains
these terms as well. Taking the ensemble average of (1/2)mq̇12
using (3) (i.e., calculating the mean kinetic energy of a
neutral particle), now gives(1/2)Θ; for the three-dimensional
motion of the particles this becomes (3/2)Θ . Ear-lier in the
paper, Lorentz has explained that one of the most important results
of kineticgas theory is that the mean kinetic energy of a moving
molecule or atom is equal to αT ,with T the temperature and α a
universal constant (not to be confused with the direc-tion cosine
introduced earlier). Thus (3/2)Θ = αT (or, in modern notation,Θ =
kT ).In this way, Lorentz extends the standard interpretation of Θ
for mechanical systemsto his much more complicated system of
particles and fields.
Now Lorentz turns to the second term in (8). Since the terms in
this sum are qua-dratic in q3, a calculation similar to the one
above shows that in an ensemble averageeach term will contribute
(1/2)Θ = (1/3)αT to the mean electric energy. Taking intoaccount
the two polarization states represented by the coordinates q3 and q
′3, we get atotal contribution by the q3’s of 2αT/3. This can also
be interpreted as the contributionof one mode (u, v, w) to the mean
electric energy.
Returning to the rectangular box, it is easy to see that the
number of modes withwavelengths between λ and λ+ dλ that fit in the
box (which is supposed to be suffi-ciently large) is equal to
4π
λ4f gh dλ. (11)
Taking into account the earlier result for the mean electric
energy per mode, Lorentzfinds
8παT
3λ4dλ (12)
for the energy density. Because the mean magnetic energy is
equal to the mean electricenergy, the total mean electromagnetic
energy density is given by twice this expression,so that the
radiation function (2) is now given by:
F(λ, T ) = 16παT3λ4
(13)
This is the Rayleigh–Jeans law.In this way, Lorentz has
generalized his earlier paper on long-wave radiation in
metals, in which he first specified a mechanism by which
radiation is generated andthen explicitly calculated the
energy-density of this radiation.
Lorentz of course realizes the consequences of his result. It is
clearly in contradic-tion with the observed radiation curve, which
shows a maximum when plotted as afunction of λ. Moreover, it
implies that in the case of a material body in equilibriumwith the
ether the latter will contain an infinite amount of energy and the
energy will
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156 A. J. Kox
become concentrated in ever shorter wavelengths.6 This had
already been concludedby Jeans. As Lorentz remarks:
[…] when Jeans published his theory, I had hoped that on closer
inspection onecould show that the theorem of equipartition of
energy on which he based him-self cannot be applied to the ether
and that in this way one could find a truemaximum of the function
F(λ, T ). The preceding considerations seem to provethat this is
not the case and that Jeans’s conclusions will be inescapable
unlessthe fundamental hypotheses of the theory are profoundly
modified.7
This is a crucial conclusion. To summarize: Lorentz has shown
that the valid-ity of the equipartition theorem for material
particles inescapably implies its valid-ity for his more
complicated mechanical–electromagnetical system (or, using hisown
words, for the ether). This then immediately leads to the
Rayleigh–Jeanslaw.8
Is there a way to reconcile the experimental results with what
he has found, Lorentzwonders. One thing is certain: for long
wavelengths the law is satisfactory; the prob-lem lies in the
short-wavelength regime. A possible solution is that the maximum
inthe observed curve is an artefact of the experiment, perhaps due
to the fact that theradiating bodies used in the experiments are
not black for small wavelengths and thusradiate much less energy
for these wavelengths than is assumed. In this way equilib-rium
between radiation and matter will take a very long time to set
in—it is in factnever observed.
In the final paragraphs Lorentz emphasizes that he does not
pretend to have providedthe definitive solution to the radiation
problem. The way one proceeds in theoreticalphysics, he argues, is
to examine the relative likelihood of various hypotheses
andtheories for a given phenomenon and study the consequences that
follow from thosehypotheses. In the case of the Planck radiation
law versus the Rayleigh–Jeans law,one has to conclude that the
latter is very hard to bring in agreement with experi-ments,
whereas Planck’s law is in good agreement with them but requires a
funda-mental change in our thinking about electromagnetic
phenomena. This is alreadyclear if one looks at a freely moving
electron emitting radiation with a continu-ous frequency spectrum.
The question remains how to apply Planck’s hypothesis ofenergy
elements in this case. Lorentz concludes by expressing the hope
that futureexperiments will provide firm evidence for one or the
other of the two radiationlaws.
6 This is what Paul Ehrenfest dubbed the ‘ultraviolet
catastrophe’.7 “[…] lorsque Jeans publia sa théorie, j’ai espéré
qu’en y regardant de plus près, on pourrait démontrer quele
théorème de l’“equipartition of energy”, sur lequel il s’était
fondé est inapplicable à l’éther, et qu’ainsion pourrait trouver un
vrai maximum de la fonction F(λ, T ). Les considérations
précédentes me semblentprouver qu’il n’en est rien et qu’on ne
pourra échapper aux conclusions de Jeans à moins qu’on ne
modifieprofondément les hypothèses fondamentales de la théorie.”8
As Einstein had already shown, given the validity of the
equipartition law, the Rayleigh–Jeans law followseven without
invoking the apparatus of statistical mechanics. See Sect. 4.2
below for details.
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Hendrik Antoon Lorentz’s struggle with quantum theory 157
4 Reactions to the Rome lecture
4.1 Wilhelm Wien
Once the Rome lecture’s contents became known among the
physicists, it stirred upquite some emotion. Wilhelm Wien wrote to
Arnold Sommerfeld on 18 May 1908:
The lecture Lorentz delivered in Rome has disappointed me
greatly. That he didnot do more than present the old theory of
Jeans without presenting a new pointof view is in my opinion a bit
shabby. […] This time Lorentz has not shownhimself as a leader of
science.9
Wien points out in rather strong terms that the question of the
validity of Jeans’slaw—or rather its non-validity—is a purely
experimental matter. He emphasizes thatLorentz’s theoretical views
on this point are irrelevant, because experiments showenormous
deviations from Jeans’s law in a region where one can easily
establish howmuch the radiating body deviates from a black
body.
Wien also communicated his objections to Lorentz himself, though
in a rather morecautious way, in a letter dated 17 May 1908:10
With much interest I have read the lecture you gave in Rome. I
think that it isvery useful to continue to keep all theoretical
possibilities in mind. But I do notthink that anyone who has ever
done experiments in the field of radiation willadmit that there is
even the remotest possibility for the theory of Jeans to
reachagreement with experience.11
He then points out that one can easily determine, by measuring
their absorptivepower, that the radiating bodies used in
experiments depart at most a few percent fromideal black bodies and
that the discrepancies of Jeans’s law with experiment are solarge
that there is no way theory and experiment can be reconciled. The
letter finisheswith an admonition of sorts:
I fear that further resistance and clinging to views that are
too simple will be animpediment to the progress of science.12
9 “Der Vortrag, den Lorentz in Rom gehalten hat, hat mich schwer
enttäuscht. Daß er weiter nichts vor-brachte als die alte Theorie
von Jeans ohne irgend einen neuen Gesichtspunkt hineinzubringen
finde ichetwas dürftig. […] Lorentz hat sich diesmal nicht als ein
Führer der Wissenschaft erwiesen.” See Eckartand Märker (2000, nr.
132).10 See Kox (2008, Letter 170).11 “Ich habe mit grossem
Interesse den Vortrag gelesen den Sie in Rom gehalten haben. Ich
glaube dasses durchaus zweckmässig ist alle theoretischen
Möglichkeiten dauernd im Auge zu behalten. Aber ichglaube nicht,
dass irgend jemand, der jemals experimentell auf dem Gebiete der
Strahlung gearbeitet hat,Ihnen zugeben wird, dass für die Theorie
von Jeans auch nur die entfernteste Möglichkeit besteht mit
derErfahrung in Übereinstimmung zu kommen.”12 “Ich fürchte daher,
dass ein längeres Sträuben und Festhalten an zu einfachen
Vorstellungen hemmendauf den Fortschritt der Wissenschaft wirken
kann.”
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158 A. J. Kox
Revisions of fundamental ideas, Wien continues, have always led
to great progressin science, provided that they forced themselves
upon us, and this is one of thoseoccasions.
Lorentz was quick to realize that he had been in error. In his
reply to Wien of 6 June190813 he admits his mistake, thanking Wien
for pointing it out to him.14 In his letterto Wien Lorentz
emphasizes that he had not meant to claim that he had proven
thecorrectness of Jeans’s law. At the same time, Wien’s letter has
convinced him that theexperimental bodies are indeed close enough
to an ideal black body to unambiguouslydisprove Jeans’s law. He
adds a very simple reasoning that makes the invalidity of thislaw
even more striking: a simple calculation shows that, were Jeans’s
law the right
13 Kox (2008, Letter 171).14 That Lorentz was quite embarrassed
about his error becomes clear from a letter to Pieter Zeeman of
20August 1908. (The letter is in the Zeeman Archive in the
Noord-Hollands Archief, Haarlem, The Nether-lands.) He wrote: “To
finish, I have to tell you that I have been very unfortunate in a
certain respect with myRome lecture. You know the theory of Jeans.
For years I had broken my head about it, and about radiationtheory
in general, and I had always some hope that, if one stayed with the
usual foundations of electrontheory, one could escape from the
theorem of “equipartition of energy” and could show that there had
to bea real maximum (inversely proportional to T ) in the radiation
function. The question whether one can applythe theorem in question
to a system consisting of ether, electrons, and atoms comes down to
whether oneis allowed to use the methods of Gibbs’s statistical
mechanics and this in turn depends on whether one canput the
equations that describe the phenomena in the form of Hamilton’s
equations of motion. When I hadfound that this is indeed the case,
it did not seem an unsuitable subject for a mathematical congress,
and soI explained there how one arrives at Jeans’s formula,
starting from the generally accepted principles. I thenalso spoke
of the well-known way in which Jeans wants to explain the
contradiction between his theoryand the observations, but
unfortunately I overlooked (I don’t quite understand how) that one
cannot enterthat road without getting into conflict with phenomena
that are very well known and simple. Thus, insteadof concluding
“only Planck’s theory is feasible”, I expressed myself at the end
as if a decision between thistheory and Jeans’s should still be
sought. W. Wien […] has written to me about this right away, and I
haveconceded the case immediately. […] I am, of course, extremely
sorry that I was wrong, but I could do nothingelse but admit it,
because spoken words, too, cannot be undone.” (“Ik moet U eindelijk
schrijven dat ik metmijne voordracht te Rome in zeker opzicht
ongelukkig ben geweest. Gij kent de theorie van Jeans. Daaroveren
over de stralingstheorie in het algemeen had ik mij jaren lang het
hoofd gebroken en ik had altijd eenigehoop dat men, als men zich
aan de gewone grondslagen der electronentheorie hield, aan het
theorema der“equipartition of energy” zou kunnen ontkomen, en zou
kunnen aantoonen dat er een werkelijk maximumder stralingsfunctie
(omgekeerd evenredig met T ) moest zijn. De vraag of men het
bedoelde theorema opeen stelsel bestaande uit aether, electronen en
atomen mag toepassen, komt hierop neer of men van demethoden der
statistische mechanica van Gibbs gebruik mag maken, en dit hangt er
weer van af of men devergelijkingen die de verschijnselen
beschrijven, in den vorm der bewegingsvergelijkingen van
Hamiltonkan brengen. Toen mij nu gebleken was dat dit inderdaad
mogelijk is, leek mij dat geen ongeschikt onder-werp voor een
mathematisch congres, en zoo heb ik daar uiteengezet hoe men
inderdaad, van de algemeenaangenomen grondbeginselen uitgaande, tot
de formule van Jeans komt. Ik heb toen ook gesproken vande bekende
wijze waarop Jeans de tegenspraak tusschen zijn theorie en de
waarnemingen wil verklaren,maar ongelukkigerwijze heb ik over het
hoofd gezien (ik begrijp zelf niet hoe) dat men dien uitweg niet
kaninslaan zonder met zeer bekende en eenvoudige verschijnselen in
strijd te komen. Dientengevolge heb ik,in plaats van te
concludeeren: “alleen de theorie van Planck is houdbaar”, mij aan
het slot van de voordrachtuitgelaten alsof er nog een beslissing
tusschen deze en die van Jeans zou gezocht moeten worden. W.
Wien[…] heeft mij er dadelijk over geschreven en ik heb hem de zaak
dadelijk toegegeven. […] Het spijt mijnatuurlijk zeer dat ik mij
vergist heb, maar ik kon niet anders doen dan het erkennen, want
ook gezegdewoorden nemen geen keer.”)
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Hendrik Antoon Lorentz’s struggle with quantum theory 159
one, a bar of silver should emit enough radiation in the visible
spectrum to be visiblein the dark at room temperature.15
Thus, Lorentz concludes, only Planck’s theory—which he claims to
admire verymuch—remains and the considerations in his Rome lecture
show that the standardelectron theory needs to be supplemented in
one way or another to solve the radiationproblem. Defending himself
against Wien’s veiled accusation of standing in the wayof the
progress of science, Lorentz stresses that he too believes that
bold new hypoth-eses lead to progress in physics and he praises the
quantum hypothesis as one of thosenew ideas. But, he adds, if we
adopt Planck’s theory, we immediately encounter aserious problem.
Elaborating on his final remarks in the Rome lecture, Lorentz
arguesas follows. If we assume that equilibrium between ether and
matter is brought aboutby Planck’s resonators, which can only
absorb or emit energy in discrete quantities,we introduce a
mechanism for which equipartition is no longer valid. But if our
sys-tem also contains free electrons, for the equilibrium between
these and the ether theequipartition law should be valid. This
means that two different equilibria exist withinone system, which
is in conflict with the second law of thermodynamics.16
4.2 Einstein
Another reaction to the Rome lecture was more positive. On 13
April 1909 AlbertEinstein wrote:
I have to tell you how much I admire the beauty of your
derivation of Jeans’s law.I cannot think of any serious objection
to this derivation. Reading your paperhas been a real event for
me.17
At first sight it seems puzzling that Einstein praised Lorentz
for his derivation ofa radiation law he was convinced to be wrong,
but if we look a little more closely atEinstein’s work we can
understand his reaction. In the paper from 1907 in which
hedeveloped his quantum theory of specific heats (Einstein 1907),
Einstein had made avery strong case for a total revision, not just
of radiation theory, but also of what hecalled ‘molecular
mechanics’. A key role in his argument was played by the
equiparti-tion law. His reasoning went as follows. Planck has
shown, on purely classical grounds,that the interaction between his
oscillators and a radiation field leads to the expression
Uν = c3
8πν2ρν (14)
15 Lorentz would use the same example in an addendum to some of
the printed versions of his Romelecture.16 Not long after the
exchange with Wien, Lorentz submitted (Lorentz 1908b), a response
to (Lummerand Pringsheim 1908), a critical paper by the
experimentalists Ernst Lummer and Otto Pringsheim. In it heretracts
his remarks on the validity of the Rayleigh–Jeans law in the Rome
lecture without any reservation.He also reiterates the argument
about two different equilibrium states first formulated in the
letter to Wien.17 “Ich muss Ihnen meine Bewunderung ausdrücken über
die Schönheit Ihrer Ableitung des Jeans’schenGesetzes. Gegen diese
Ableitung wüsste ich keinen ernst zu nehmenden Einwand. Die Lektüre
Ihrer Ab-handlung ist für mich ein Ereignis.” See Kox (2008, Letter
185).
123
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160 A. J. Kox
with Uν the mean energy of an oscillator with frequency ν and ρν
the energy density ofradiation of the same frequency. Assuming that
equipartition holds for the oscillators,we have U = kT .
Substituted in (14) this gives the Jeans radiation law, which,
asEinstein points out, is only valid for large values of T/ν.
Einstein’s conclusion is thateven for a system of oscillators we
have to modify the equipartition law because itleads to serious
contradictions.18 So, what Lorentz had done was to provide in
effecta much stronger basis for the argument that equipartition was
the problem, and thismust have pleased Einstein.19
In Lorentz’s reply, a lengthy letter dated 6 May 1909,20 as well
as in a letter to Wienfrom 12 April 190921 it becomes clear that
his thinking about the quantum problemhas evolved and that he now
accepts the need for energy elements:
I no longer have doubts that only with the help of the
hypothesis of energyelements one can arrive at the correct
radiation-law.22
In his reply to Einstein Lorentz also admits that one cannot do
without energyelements. He reiterates some of the points already
discussed in his earlier correspon-dence with Wien and Planck: the
problem of the existence of two equilibrium statesin a system
containing free electrons as well as oscillators and the question
of whereto localize the discontinuity.
Lorentz uses the occasion of his letter to Einstein to discuss
another importantquantum issue that had only been touched upon
briefly in his earlier correspondence:how to deal with the light
quanta, postulated by Einstein in 1905:
I find it hard to subscribe to the view that light quanta retain
a certain individualityeven during their propagation, as if one
were dealing with point-like quantitiesof energy or at least energy
quantities concentrated in very small spaces.23
He now works out his objections in detail, emphasizing in
particular the problems alocalized light quantum poses in
explaining interference and the resolving power oftelescopes. He
concludes that a light quantum must have a length of at least
dozens ofcentimeters in order to account for the observed
possibility of interference with phasedifferences of millions of
wavelengths. From the fact that the resolving power of atelescope
gets better with increasing aperture, he then infers that light
quanta shouldhave a larger lateral extension than the aperture of
any telescope. For an aperture of,say, 50 cm this leads to an
estimate for the lateral extension of at least 5,000 cm2
18 He then proceeds to show how one can modify the structure of
phase space, namely by quantizing it, toobtain Planck’s law from a
statistical mechanical calculation.19 One might wonder why Lorentz
did not refer to Einstein’s 1907 paper in his Rome lecture.20 Kox
(2008, Letter 189).21 Kox (2008, Letter 184).22 “Ich zweifle jetzt
gar nicht mehr daran, dass man nur mit Planck’s Hypothese der
Energieelemente (dieman übrigens noch in verschiedener Weise
auffassen kann) zu der richtigen Strahlungsformel gelangt.”Lorentz
to Wien, 12 April 1909.23 “Ich kann mich aber schwerlich der
Meinung anschliessen, dass die Lichtquanten auch während
derFortpflanzung eine gewisse Individualität behalten, als ob man
es mit ‘punktförmigen’ oder jedenfalls insehr kleinen Räumen
konzentrierten Energiemengen zu tun hätte.”
123
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Hendrik Antoon Lorentz’s struggle with quantum theory 161
(because light quanta that do not hit the telescope right in the
middle of the objectiveshould still cover the full opening).
Lorentz would raise these objections time andagain in later papers
as well as in correspondence with colleagues.
Einstein was pleased with Lorentz’s letter. To his former
collaborator Jakob Laubhe wrote on 19 May 1909:
I am presently carrying on an extremely interesting
correspondence with H. A.Lorentz on the radiation problem. I admire
this man like no other: I might evensay, I love him.24
Einstein takes the objections raised by Lorentz very seriously,
emphasizing not somuch their points of disagreement but rather
where they agree. He suggests that thediscrepancy between the
behavior of free electrons and resonators should be resolvedby a
suitable generalization of Planck’s hypothesis, which is simply too
narrow if onlyapplied to monochromatic resonators. He also denies
that he subscribes to the idea ofdiscrete, independent point-like
light quanta, but instead suggests a picture in whichthese quanta
are singularities that are surrounded by a vector field whose
strengthdecreases with increasing distance. The field energy is
then related somehow to thenumber of these singularities.
4.3 An objection by Van der Waals Jr.
A more technical problem with Lorentz’s approach in the Rome
lecture was pointedout by Johannes Diderik van der Waals Jr. (at
the time professor at the University ofGroningen but soon to be his
famous father’s successor in Amsterdam). He objectedthat Lorentz’s
formalism only works when one assumes that the electrons have
materialmass, in addition to their electromagnetic mass.25 This was
a very serious assump-tion, because a consensus had more or less
been reached that electrons only possessan electromagnetic mass.
Especially Walther Kaufmann’s experiments on the ratio ofcharge to
mass of electrons seemed to indicate that they have no material
mass.26
Van der Waals’s reasoning went as follows: if the electrons lack
mechanical mass,accelerations no longer occur in the equations of
motion. This circumstance leads torelations between the coordinates
q2 and q̇2, so that these are no longer independent.Because of this
a canonical ensemble cannot be formed, so that the whole
edifice
24 “Mit H.A. Lorentz habe ich gegenwärtig eine überaus
interessante Korrespondenz über das Strahlungs-problem. Ich
bewundere diesen Mann wie keinen andern, ich möchte sagen, ich
liebe ihn.” (Klein et al.1993, Doc. 161).25 See Van der Waals
(1909).26 See, e.g., Miller (1981, chapter 1.11), for a discussion
of these experiments. See also the followingstatement by Lorentz in
section 32 of Lorentz (1909b), his 1906 lectures at Columbia
University: “[…]with a view to simplicity, it will be best to admit
Kaufmann’s conclusion that the negative electrons haveno material
mass at all.” And in section 34 he generalizes his view in the
following way: “I for one shouldbe quite willing to adopt an
electromagnetic theory of matter and of the forces between material
particles.[…] Therefore, […] all forces may be regarded as
connected more or less intimately with those which westudy in
electromagnetism.” These forces, Lorentz argues, include molecular
forces and gravitation. SeeMcCormmach (1970) for more on this
‘electromagnetic world picture.’
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162 A. J. Kox
of Lorentz’s reasoning collapses.27 That Lorentz took this
objection very seriouslybecomes clear from his correspondence. He
mentions the objection in letters to Ein-stein28 and to Arnold
Sommerfeld,29 adding in both cases that he has been unable tofind a
solution—except the obvious one of postulating that there is a
material mass, butthat it is vanishingly small. The most detailed
discussion is found in correspondencewith Van der Waals himself, in
particular in a letter of 7 April 1909. Here Lorentzintroduces the
assumption that in the system under consideration the speeds of
theelectrons are small as compared to the speed of light. For
vibrating electrons, thismeans that their amplitudes are small
compared to the wavelength of the radiation andfor non-vibrating
ones that their state of motion changes noticeably over distances
thatare small with respect to the wavelength of the radiation.
Thus, Lorentz’s assumptiononce again amounts to a condition on the
wavelengths allowed in the system under con-sideration. On his
assumption, Lorentz shows, the coordinates become independentagain.
Still, he must have felt uncertain about his supplementary
condition, becausehe did not follow up on his plans to publish his
ideas, in spite of an announcement inthe letter to Van der Waals
that he would do so. Only in 1911, in his lecture at the
firstSolvay Conference did Lorentz openly speak out on this point
(see Sect. 5).
It is interesting that in the correspondence with Van der Waals,
nor in his Solvaylecture does Lorentz counter Van der Waals’s
objection by simply introducing a smallmaterial mass for the
electrons and working out the consequences. In a letter to Vander
Waals of 19 November 1908 Lorentz does raise this possibility, but
he then rejectsit. His argument is that unforeseen problems might
arise in the limiting case of zeromaterial mass, which one would
have to consider if only electromagnetic mass shouldexist—and, as
we saw, Lorentz believed this to be the case. As he puts it: “In
any case,it is much safer to directly consider electrons without
material mass.”30
5 The first Solvay Conference
The first Solvay Conference, which was held in Brussels from 30
October to 3 Novem-ber 1911, offered Lorentz the opportunity to
express himself on the crisis in physicsthat the emerging quantum
theory had caused. In his opening address as chairman ofthe meeting
he showed himself far from optimistic:
At the moment we have the feeling that we are at a dead end,
with the old the-ories showing themselves more and more powerless
to pierce the darkness thatsurrounds us from all sides. […] What
will be the result of these meetings? Idare not predict it, not
knowing what surprises may be in store for us. But, as
27 If in phase space the coordinates and their time derivatives
are no longer independent, the incompress-ibility condition
∑∂ ṡi /∂si , with si the coordinates of the phase space, is no
longer satisfied, so that the
ensemble is not stationary.28 Lorentz to Einstein, 6 May 1909
(Kox 2008, Letter 189).29 Lorentz to Sommerfeld, 23 November 1908
(Kox 2008, Appendix, Letter 178a).30 “In elk geval is het veel
veiliger, rechtstreeks electronen zonder materieele massa te
behandelen.”
123
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Hendrik Antoon Lorentz’s struggle with quantum theory 163
it is wise not to count on surprises, I admit that it is very
likely that we willcontribute only little to the immediate
progress.31
Lorentz’s lecture at the Solvay meeting (Lorentz 1912b), dealt
with the same sub-ject as his Rome lecture, but its main point, the
problem of the validity of the theoremof equipartition of energy in
the theory of radiation, now appears already in its title.Lorentz
accepts the failure of classical theory to account for the
radiation experimentsfrom the outset and points out how Planck’s
hypothesis of energy elements has beenmore successful and has even
found “unexpected verifications” (“verifications inatten-dues”).
But he finds it still useful to reiterate his Rome derivation of
the Rayleigh–Jeanslaw to bring out clearly what the problems are
with the classical approach. As he puts it:
Before starting the discussion of Planck’s hypothesis, it is
perhaps useful tobecome aware of the shortcomings of the old
theories.
Lorentz’s approach is now more systematic: he first reproduces
the simple deriva-tion of the Rayleigh–Jeans law on the basis of
the equipartition law that was earliergiven by Einstein (see Sect.
4.2), and then immediately poses his central question:
Is there a way to escape from the equipartition theorem, either
in general, or inin its application to the problem that we are
occupied with?32
Lorentz then essentially repeats the calculation of the Rome
lecture, although withone important difference: whereas in the Rome
lecture Lorentz glosses over his intro-duction of the “fictive
connections” that put a lower limit on the allowed wavelengths,and
in fact proceeds as if this limit does not exist, he now emphasizes
this cutoff asessential to avoid the problem of zero material mass
of the electrons. He does add, how-ever, that the restriction on
the possible wavelengths does not resolve the discrepancybetween
his outcome (the Rayleigh–Jeans law) and experiments, as the
experimentaldifferences show themselves also at larger wavelengths
than the extremely small onesexcluded here. In the end, however,
the conclusion of the Solvay lecture is the sameas the one drawn in
the Rome lecture: the equipartition theorem is incompatible withthe
observed form of the radiation curve. Obviously, after the critical
reactions to theRome lecture, Lorentz no longer tries to find a way
out by doubting the validity of theradiation measurements.
6 The discontinuity
In addition to the problem of equipartition of energy and the
validity of theRayleigh–Jeans law, another important point of often
confused discussion emerged inthe correspondence with Wien, Planck
and Einstein following the Rome lecture. It is
31 “Nous avons maintenant le sentiment de nous trouver dans une
impasse, les anciennes théories s’étantmontrés de plus en plus
impuissantes à percer les ténèbres qui nous entourent de tous
côtés. […] Quel serale résultat de ces réunions? J n’ose le
prédire, ne sachant pas quelles surprises peuvent nous être
réservées.Mais, comme il est prudent de ne pas compter sur les
surprises, j’admettrai comme très probable que nouscontribuerons
pour peu de chose au progrès immédiat.” (Lorentz 1912a, pp. 7–8).32
“Y a-t-il moyen d’échapper, soit au théorème de l’équipartition en
general, soit à son application auproblème qui nous occupe?”
123
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164 A. J. Kox
the question where one should localize the discontinuity—if
there is one. In a letterto Planck, for instance, Lorentz maintains
that processes that take place in the etherare continuous in
character, agreeing with Planck on this point.33 Two months
later,he admits that one cannot do without light quanta, but
immediately adds his standardobjections against the individuality
of these quanta. He now claims that h might bea constant of the
ether: the Maxwell equations are valid, but links between groups
ofvibrations limit the number of degrees of freedom of the ether, a
limitation that finds itsexpression in the occurrence of Planck’s
constant.34 Two months later Lorentz spec-ulates again that h has
something to do with the particles that produce radiation.35
His final view, expressed in 1912, seems to be that the
interaction between ether andresonators is continuous, whereas the
energy exchange between ordinary matter, suchas atoms, and
resonators is somehow quantized.36
7 The light quantum
After his initial discussion with Einstein (see Sect. 4.2)
Lorentz kept arguing againstthe independent existence of light
quanta in his correspondence and in later publi-cations. He was of
course not alone in his objections: in fact, the large majority
ofphysicists rejected the existence of light quanta on grounds
similar to the ones putforward by Lorentz.37
Let me briefly mention two later publications in which Lorentz
expresses himselfstrongly on this point. The first is a paper from
1909, with the title “The hypothe-sis of light quanta” (Lorentz
1909a). Here Lorentz first gives an elegant derivation ofPlanck’s
law, using the now well-known combinatorial expression to count the
numberof ways to distribute p identical elements over n
indistinguishable resonators. He thendiscusses the light quantum
hypothesis and its successful application to phenomenasuch as
Stokes’s law for phosphorescence38 and the photoelectric effect.39
He alsodiscusses experiments by Stark on canal rays that seem to
support the light quantumhypothesis. The discussion is, as always,
fair and thorough, but he concludes:
33 Lorentz’s letter is lost, but its contents may be partially
reconstructed from Planck’s reply of 24 March1909 (Kox 2008, Letter
187).34 See Planck’s summary in Planck to Lorentz, 16 June 1909
(Kox 2008, Letter 192).35 See Kox (2008, Letters 194 and 195)
(drafts for a letter dated 30 July 1909, now lost).36 See Lorentz
(1912c). In this paper Lorentz tries to explore in a purely
classical way the consequences ofthe hypothesis that somehow during
collisions between resonators in a solid and the atoms of a
surroundinggas energy is only exchanged in discrete quanta. He is
motivated by Einstein’s successful theory of specificheats, in
which quantized monochromatic oscillators are the building blocks
of solids.37 Typical is the assessment given by Planck, Nernst,
Rubens and Warburg in their proposal to award Ein-stein a salaried
membership of the Prussian Academy of Sciences: “That he might
sometimes have overshothis target in his speculations, as for
example in his light quantum hypothesis, should not be counted
againsthim too much.” (“Daß er in seinen Spekulationen gelegentlich
auch einmal über das Ziel hinausgeschossenhaben mag, wie z.B. in
seiner Hypothese der Lichtquanten, wird man ihm nicht allzuschwer
anrechnendürfen.”) (Klein et al. 1993, Doc. 445).38 The light
emitted in a phosphorescence process is always lower in frequency
than the light absorbed.39 These two processes were also discussed
by Einstein as evidence for his light quantum hypothesis inEinstein
(1905).
123
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Hendrik Antoon Lorentz’s struggle with quantum theory 165
This is all without doubt very striking, but nevertheless it
seems to me that oncloser inspection grave objections to the light
quantum hypothesis arise.40
Lorentz then reiterates his reasoning about the size of the
quanta and the paper endswith the conclusion:
What has been said may be sufficient to show that there can be
no question of lightquanta that remain concentrated in small spaces
and remain always undividedduring their propagation.41
A few years later, during a discussion at the September 1913
meeting of the Brit-ish Association for the Advancement of Science,
Lorentz repeats his rejection of theexistence of light quanta:
Now it must, I think, be taken for granted, that the quanta can
have no individualand permanent existence in the ether, that they
cannot be regarded as accumu-lations of energy in certain minute
spaces flying about with the speed of light(Lorentz 1913, p.
381).
Eventually, though, Lorentz had to modify his views, especially
once Einstein’sexplanation of the photoelectric effect had been
confirmed by Robert Millikan’s exper-iments. But he remained
concerned because the problem of how to reconcile the exis-tence of
light quanta with phenomena like interference remained
unsolved.
In 1921 Lorentz finally saw a way out through an idea first
formulated by Einsteinin a discussion they had in Leiden. Einstein
never published his idea, but he refers toit obliquely in a paper
from 1921.42
The first elaboration of Einstein’s idea came in a letter from
Lorentz to Einsteinof 13 November 1921, written by Lorentz to make
sure he had understood Einsteincorrectly.43 The mechanism he
outlines is clearly inspired by De Broglie’s postulatedpilot waves:
whenever radiation is emitted, this radiation consists of two
parts, whichLorentz calls energy radiation and interference
radiation. The latter carries no energy,but has a wavelike
character (one might think of ordinary electromagnetic waves
withinfinitesimally small amplitude). It paves the way, so to
speak, for the quantized energyradiation that follows it. The idea
is that the light quanta are no longer completely freein their
motion; where they can go, and how many can go to a certain spot is
dictatedby the ‘intensity’ of the interference radiation. Let us
take the double-split experimentas an example. The interference
radiation creates the well-known interference patternon the screen
behind the slits, but because the radiation carries no energy we
cannotsee it. The number of light quanta that land on the screen is
determined by this inter-ference pattern: the higher the
‘intensity’ in a spot on the screen, the more light quanta
40 “Dies alles ist ohne Zweifel sehr auffallend, aber trotzdem
will mich dünken, dass bei näherer Betrach-tung ernste Bedenken
gegen die Hypothese der Lichtquanten aufsteigen.”41 “Das Gesagte
dürfte genügen, um zu zeigen, dass von Lichtquanten, die bei der
Fortbewegung in kleinenRäumen konzentriert und stets ungeteilt
bleiben, keine Rede sein kann.”42 See Einstein (1921). In this
paper Einstein proposes an experiment to determine whether canal
rays havea wave-like or a particle-like structure. The reasoning
behind the experiment turned out to be flawed: seeJanssen et al.
(2002, Doc. 68, note 5).43 Kox (2008, Letter 371).
123
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166 A. J. Kox
reach that spot. Where the ‘intensity’ is zero, no light quanta
arrive. Thus the familiarinterference pattern is made visible by a
succession of individual quanta.
The Einstein–Lorentz idea had little resonance with other
physicists. But Lorentzremained charmed by it: he included a more
elaborate form, including the interac-tion of two beams, in
lectures given in 1922 at the California Institute of
Technology(Caltech),44 as well as in lectures at Cornell University
in the fall of 1926.45
8 Wave mechanics and matrix mechanics
While in the years after 1911 quantum physics made huge
progress, with the break-through achieved by Bohr in 1913 and the
subsequent development of what is nowknown as the ‘old quantum
theory’ by Sommerfeld and others, Lorentz remained skep-tical. As
late as 1925 in a lecture at the Société Française de Physique he
summarizedhis misgivings in the following way:
All this [i.e. quantum theory] is of great beauty and
importance, but unfortu-nately we do not understand it. We do not
understand Planck’s hypothesis on theoscillators, nor do we
understand the exclusion of non-stationary orbits and wedo not see
how in Bohr’s theory the light is eventually produced. For,
admittedly,the mechanics of quanta, the mechanics of
discontinuities, still has to be made.46
It would take another 2 years before the first steps were taken
towards a true quan-tum mechanics. At the end of 1925 Werner
Heisenberg published the ground-breakingpaper in which he developed
the formalism of matrix mechanics and in the first monthsof 1926
Erwin Schrödinger published his wave mechanics.47 It is interesting
to com-pare Lorentz’s reactions to the two new approaches. From his
correspondence, inparticular with Paul Ehrenfest, it becomes clear
that he studied Heisenberg’s originalpaper and the further
development of matrix mechanics by Heisenberg, Born, Jordan,and
Dirac. He even lectured in Leiden on matrix mechanics, as early as
the fall of1926.48 But he never engaged with matrix mechanics in
the way he did with wavemechanics. There is no correspondence on
technical points with the authors men-tioned earlier, and in his
correspondence with Ehrenfest, for example, Lorentz seeksout
Ehrenfest’s help and opinion rather than trying to extend the
theory.
44 See Lorentz (1927, secs. 50–53).45 A set of mimeographed
lecture notes of the Cornell lectures is preserved in the Caltech
Archives.46 “Tout cela est d’une grande beauté et d’une extrême
importance, mais malheureusement nous ne lecomprenons pas. Nous ne
comprenons ni l’hypothèse de Planck sur les vibrateurs, ni
l’exclusion des orbitesnon stationnaires et nous ne voyons pas,
dans la théorie de Bohr, comment, en fin de compte, la lumière
estproduite. Car, il faut bien l’avouer, la mécanique des quanta,
la méchanique des discontinuités, doit encoreêtre faite.” (Lorentz
1925).47 See Heisenberg (1925) and Schrödinger (1926a,b); see also,
for instance, Jammer (1966) or Mehra andRechenberg (1982–2001,
vols. 2 and 5), for historical overviews of the development of
matrix mechanicsand wave mechanics.48 See his lecture notes in nrs.
289, 574, and 576 in the Lorentz Archive in the Noord-Hollands
Archief,Haarlem, The Netherlands.
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Hendrik Antoon Lorentz’s struggle with quantum theory 167
With wave mechanics the situation was different. On 30 March
1926 Schrödingersent Lorentz the proofs of his first two papers on
wave mechanics (Schrödinger1926a,b), asking Lorentz for his
comments.49 On 27 May the 72-year-old Lorentzreplies with a letter
of nine densely written pages,50 from which it becomes clear thathe
has thoroughly analyzed Schrödinger’s papers. Not surprisingly,
Schrödinger’sapproach appealed to him, as being more ‘anschaulich’
than the much more abstractmatrix mechanics. Lorentz writes that he
had very much enjoyed studying the papers,but that he has also
found some problems that, in his view, might be unsurmountable:
– It will be very difficult to give a physical interpretation of
Schrödinger’s wave func-tions ψ , because they are complex
quantities in a high-dimensional configurationspace;
– A calculation has shown that wave packets formed from such
wave functions can-not represent stable particles, because of their
rapid spreading.
These problems cast doubt in particular on one of the basic
ideas behind wavemechanics: the extension of Hamilton’s old analogy
between mechanics and geomet-rical optics.51 As Lorentz put it:
Your idea that the change which our dynamics must undergo will
be similar tothe transition from geometrical optics to wave optics
sounds very enticing, butI have doubts about it.52
In his reply53 Schrödinger tries to counter Lorentz’s
objections. He suggests thatone needs to look atψ∗ψ (withψ∗ the
complex conjugate) to find a physical interpre-tation of the wave
function and suggests that this quantity is related to the
electricalcharge density.54 He also refers to a note that he has
included and in which he showsthat that for a harmonic oscillator
stable wave packets can be constructed, indicat-ing that perhaps
there is still hope for free or bound electrons to be represented
bypackets.55
49 See Kox (2008, Letter 405). Schrödinger later also sent the
proofs of his third paper, in which he showsthe formal equivalence
of wave mechanics and matrix mechanics; see ibid. Letter 412.50 Kox
(2008, Letter 412).51 In short: Hamilton had shown that there is a
formal analogy between mechanics and geometrical optics,in the
sense that the Hamilton–Jacobi formalization of mechanics can be
translated into the eikonal for-malism of geometrical optics. Since
geometrical optics is the short-wavelength limit of wave optics,
onemight wonder whether classical mechanics is the limiting case of
some wave theory which is analogousto wave optics. In Hamilton’s
days there was no reason to suppose that mechanics might have any
kind ofwave-like character, but when Schrödinger was looking for a
quantum theoretical generalization of classicalmechanics, trying to
find a wave-like theory was a plausible way to proceed. In fact,
his derivation of theSchrödinger equation in Schrödinger (1926b)
takes this approach. See also Goldstein (1950) for more onthe
optical–mechanical analogy and Joas and Lehner (2009) for
Schrödinger’s inspiration by the analogy.52 “Ihre Vermutung, dass
die Umwandlung, welche unsere Dynamik wird erfahren müssen, dem
Übergangevon Strahlenoptik zu Wellenoptik ähnlich sein wird, klingt
sehr verlockend, aber ich habe doch Bedenkendagegen.”53 Kox (2008,
Letter 413).54 Not long afterwards Schrödinger had to abandon this
interpretation.55 The included note is a copy of the manuscript of
Schrödinger (1926b).
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168 A. J. Kox
Lorentz replied as quickly as 19 June with a letter of 19 large
pages full of calcula-tions.56 He essentially crushes Schrödinger’s
hopes with a new and detailed analysis ofthe behavior of wave
packets in wave mechanics. In his usual gentle way he concludesthat
there is little hope left for Schrödinger’s theory:
If we have to give up the wave packets and with it one of the
fundamental ideasof your theory, the transformation from classical
mechanics to a wave mechan-ics, something would be lost that would
have been very beautiful. I would beextremely pleased if you could
find a way out.57
Lorentz believed to have shown Schrödinger’s theory to be
untenable—at least froma purely classical point of view. As we
know, Schrödinger did not give up. Completenew interpretations of
the wave equation and the wave function were needed to givephysical
meaning to the theory. But Lorentz had gone as far as he could go:
reachingfurther beyond the boundaries of classical physics was too
much for him.58
9 Conclusion
From the discussions in the preceding sections a few key points
have emerged, bothconcerning Lorentz’s concrete contributions and
his approach to quantum theory:
– Lorentz’s way of thinking about the radiation problem was
strongly influenced byhis views on the constitution of matter and
on the interaction between matter andether, views that had led to
his eminently successful electron theory.
– The conclusion of his Rome lecture that classical theory
necessarily leads to theRayleigh–Jeans law and that the discrepancy
between the experimental radiationcurve and classical theory can
only be removed by a radical new element has servedas an important
clarification of the radiation problem, in particular about the
roleof the equipartition theorem.
– It is fair to say that, after the clarification provided by
the Rome lecture, Lorentz didnot come up with any new ideas but
kept repeating his objections. He proceeded ashe always did in
physics: with great caution, and with impressive technical
mastery.In this particular case he was perhaps more critical and
cautious than in his earlier,purely classical work, because of his
desire to keep as much of classical theoryintact as possible. In
specific cases, such as the discussion with Schrödinger, thisled to
useful clarifications, but real progress in quantum theory had to
come fromrepresentatives of a younger generation, who embraced the
quantum hypothesis assomething new and unavoidable and made daring
applications of it.
56 Kox (2008, Letter 416).57 “Indes, wenn wir die Wellenpakete
aufgeben müssen und damit einen der Grundgedanken Ihrer Theorie,die
Umwandlung der klassischen Mechanik in eine undulatorische, so
würde damit etwas verloren gehen,das sehr schön gewesen wäre. Es
würde mich sehr freuen, wenn Sie hier einen Ausweg finden
könnten.”58 In spite of his misgivings, Lorentz lectured on wave
mechanics (as well as on matrix mechanics) whilevisiting Cornell
University in the fall of 1926 and the California Institute of
Technology in the first monthsof 1927. See Footnote 45; see also
Lorentz to Schrödinger, 21 January 1927 (Kox 2008, Letter 420),
forevidence that Lorentz covered the same material at both
institutions.
123
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Hendrik Antoon Lorentz’s struggle with quantum theory 169
– To the very end, and true to his status as classical physicist
par excellence, Lorentzkept thinking in purely classical terms and
tried to adhere as closely as possible toclassical theory in his
work on quantum theory.59
As will have become clear from the preceding discussion, Lorentz
also remainedtrue to himself in another more general way: in his
capacity to objectively evaluate andappreciate points of view of
others, without giving up his own, strong convictions.60
Although he made no secret of his personal preferences for
specific approaches, henever rejected alternatives out of hand and
was willing and able to accept new con-cepts, such as the light
quantum, when the evidence was overwhelming. Still, in theend, he
was and remained a classical physicist.
Acknowledgments I am much indebted to Jed Buchwald for his
critical comments on an earlier versionof this paper and to
Henriette Schatz for valuable suggestions.
Open Access This article is distributed under the terms of the
Creative Commons Attribution Licensewhich permits any use,
distribution, and reproduction in any medium, provided the original
author(s) andthe source are credited.
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Hendrik Antoon Lorentz's struggle with quantum theoryAbstract1
Introduction2 Early work on radiation theory3 The Rome lecture4
Reactions to the Rome lecture4.1 Wilhelm Wien4.2 Einstein4.3 An
objection by Van der Waals Jr.
5 The first Solvay Conference6 The discontinuity7 The light
quantum8 Wave mechanics and matrix mechanics9
ConclusionAcknowledgmentsReferences